I can't seem to recreate this - works fine here. Have you cleared your browser cache lately? We updated chat but that was 3-4 months ago now.
I see the same problem. Right now in the forum it says online, but on the homepage it shows offline.What i see a lot, is that in some pages i see the chat "Offline" and in others "online" it's like random
Edit: i have opened at the same time, the forum and the homepage, and i see chat online at the forum, but offline at the homepage.
Edit2: moving from the homepage, to a section, shows the chat online again (coming back to the homepage remains the chat online).